Measures of Variability
What is Variability?
Variability refers to how "spread out" a group of scores is. To see
what we mean by spread out, consider graphs in Figure 1. These
graphs represent the scores on two quizzes. The mean score for
each quiz is 7.0. Despite the equality of means, you can see that
the distributions are quite different. Specifically, the scores on Quiz
1 are more densely packed and those on Quiz 2 are more spread
out. The differences among students was much greater on Quiz 2
than on Quiz 1.
Figure 1. Bar charts of two quizzes.
Quiz 1
Quiz 2
The terms variability, spread, and dispersion are synonyms, and
refer to how spread out a distribution is. Just as in the section on
central tendency we discussed measures of the center of a
distribution of scores, in this chapter we will discuss measures of
the variability of a distribution. There are four frequently used
measures of variability, the range: interquartile range, variance,
and standard deviation. In the next few paragraphs, we will look at
each of these four measures of variability in more detail.
Range
The range is the simplest measure of variability to calculate, and
one you have probably encountered many times in your life. The
range is simply the highest score minus the lowest score. Let’s take
a few examples. What is the range of the following group of
numbers: 10, 2, 5, 6, 7, 3, 4? Well, the highest number is 10, and
the lowest number is 2, so 10 - 2 = 8. The range is 8. Let’s take
another example. Here’s a dataset with 10 numbers: 99, 45, 23,
67, 45, 91, 82, 78, 62, 51. What is the range? The highest number
is 99 and the lowest number is 23, so 99 - 23 equals 76; the range
is 76. Now consider the two quizzes shown in Figure 1. On Quiz 1,
the lowest score is 5 and the highest score is 9. Therefore, the
range is 4. The range on Quiz 2 was larger: the lowest score was 4
and the highest score was 10. Therefore the range is 6.
Interquartile Range
The interquartile range (IQR) is the range of the middle 50% of the
scores in a distribution. It is computed as follows:
IQR = 75th percentile - 25th percentile
For Quiz 1, the 75th percentile is 8 and the 25th percentile is 6. The
interquartile range is therefore 2. For Quiz 2, which has greater
spread, the 75th percentile is 9, the 25th percentile is 5, and the
interquartile range is 4. Recall that in the discussion of box plots,
the 75th percentile was called the upper hinge and the 25th
percentile was called the lower hinge. Using this terminology, the
interquartile range is referred to as the H-spread.
A related measure of variability is called the semi-interquartile
range. The semi-interquartile range is defined simply as the
interquartile range divided by 2. If a distribution is symmetric, the
median plus or minus the semi-interquartile range contains half the
scores in the distribution.
Variance
Variability can also be defined in terms of how close the scores in
the distribution are to the middle of the distribution. Using the mean
as the measure of the middle of the distribution, the variance is
defined as the average squared difference of the scores from the
mean. The data from Quiz 1 are shown in Table 1. The mean score
is 7.0. Therefore, the column "Deviation from Mean" contains the
score minus 7. The column "Squared Deviation" is simply the
previous column squares.
One thing that is important to notice is that the mean deviation
from the mean is 0. This will always be the case. The mean of the
squared deviations is 1.5. Therefore, the variance is 1.5. Analogous
calculations with Quiz 2 show that it's variance is 6.7. The formula
for the variance is:
where σ2 is the variance, μ is the mean, and N is the number of
numbers. For Quiz 1, μ = 7 and N = 20.
If the variance in a sample is used to estimate the variance in a
population, then the previous formula underestimates the variance
and the following formula should be used:
where s2 is the estimate of the variance and M is the sample mean.
Note that M is the mean of a sample taken from a population with a
mean of μ. Since, in practice, the variance is usually computed in a
sample, this formula is most often used. The simulation "estimating
variance" illustrates the bias in the formula with N in the
denominator.
Table 1. Calculation of Variance for Quiz 1 scores.
Scores
Deviation
from Mean
Squared
Deviation
9
2
4
9
2
4
9
2
4
8
1
1
8
1
1
8
1
1
8
1
1
7
0
0
7
0
0
7
0
0
7
0
0
7
0
0
6
-1
1
6
-1
1
6
-1
1
6
-1
1
6
-1
1
6
-1
1
5
-2
4
5
-2
4
7
0
1.5
Let's take a concrete example. Assume the scores 1, 2, 4, and 5
were sampled from a larger population. To estimate the variance in
the population you would compute s2 as follows:
M = (1 + 2 + 4 + 5)/4 = 12/4 = 3.
s2 = [(1-3)2 + (2-3)2 + (4-3)2 + (5-3)2]/(4-1)
= (4 + 1 + 1 + 4)/3 = 10/3 = 3.333
There are alternate formulas that can be easier to use if you are
doing your calculations with a hand calculator:
and
For this example,
ΣX2 = 12 + 22 + 42 + 52 = 46
(ΣX)2 = (1 + 2 + 4 + 5)2/N = 144/4 = 36
σ2 = (46 - 36)/4 = 2.5 and
s2 = (46 - 36)/3 = 3.333 as with the other formula.
Standard Deviation
The standard deviation is simply the square root of the variance.
This makes the standard deviations of the two quiz distributions
1.225 and 2.588. The standard deviation is an especially useful
measure of variability when the distribution is normal or
approximately normal because the proportion of the distribution
within a given number of standard deviations from the mean can be
calculated. For example, 68% of the distribution is within one
standard deviation of the mean and approximately 95% of the
distribution is within two standard deviations of the mean.
Therefore, if you had a normal distribution with a mean of 50 and a
standard deviation of 10, then 68% of the distribution would be
between 50 - 10 = 40 and 50 +10 =60. Similarly, about 95% of the
distribution would be between 50 - 2 x 10 = 30 and 50 + 2 x 10 =
70. The symbol for the population standard deviation is σ; the
symbol for an estimate computed in a sample is s. Figure 2 shows
two normal distributions. Both distributions have means of 50. The
blue distribution has a standard deviation of 5; the red distribution
has a standard deviation of 10. For the blue distribution, 68% of the
distribution is between 45 and 55; for the red distribution, 68% is
between 40 and 60.
Figure 2. Normal distributions with standard
deviations of 5 and 10.